Continued fraction expansion

The LDOS have been calculated using 10 exact levels in the continued fraction expansion of the Green functions. For clean surfaces the quantities A Vi are the same for all atoms in the same plane they have been determined up to p = 2, 4, 6 for the (110), (100) and (111) surfaces, respectively, and neglected beyond. The cluster C includes the atoms located at the site occupied by the impurity and at al the neighbouring sites up to the fourth nearest neighbour. 

F(q, t) is obtained from its Laplace transform form F(q,z)- By using the following well-known Mori continued-fraction expansion and truncating at second order, the viscoelastic expression for F(q, z) can be written as [16, 21, 22]... 

Historically this number played an important role in ancient Greek geometry and in concepts of aesthetics and beauty. It is also a key number in chaos theory. In mathematics the significance of g lies in the fact that it is the most irrational number . Its continued fraction expansion is ... 

By exploiting the diffusional form of Eq. (5.18), which allows an analytical expression for numerical algorithm based on a continued fraction expansion (Grosso and Pastori Parravicini, Chapter III), we can compute the correlation function ( c(/)a (0))o. Here ( >o means an average over the equilibrium realizations accounted by a ix). In this case the assumption we made to get rid of the inhomogeneous term of Eq. (3.5) should now read ... 

All the mathematical apparatus of Hankel determinants and continued fractions expansion apply also to Hermitian or relaxation superoperators. 

The procedure can be iterated to obtain the familiar continued-fraction expansion of the Green s function Goq( ) ... 

From Eqs. (3.46) we arrive thus at the continued fraction expansion... 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

All the relationships of the present section maintain their validity provided that the (/j are systematically replaced by (. In particular, the parameters a and of the continued fraction expansion (3.48) are now given by [compare with Eqs. (3.41) and (3.42)]... 

As an example of the study of vacancies and self-interstitial impurities by the continued fraction expansion of Eq. (S.2S), we mention the work of Kauffer et al. These authors consider impurities in silicon and set up a model tight-binding Hamiltonian with s p hybridization, which satisfactorily describes the valence and conduction bands of the perfect crystal. A cluster of 2545 atoms is generated, and vacancies (or self-interstitial impurities) are introduced at the center of the cluster. One then takes as a seed state an appropriate orbital or symmetrized combination of orbitals, and the recursion method is started. Though self-consistent potential modifications are neglected in this paper, the model leads to qualitatively satisfactory results within a simple physical picture. 

The work of other authors cannot be clearly classified as belonging to one of these three main families. A quantum-statistical theory of longitudinal magnetic relaxation based on a continued fraction expansion has been given by Sauermann, who also pointed out the equivalence between his continued fraction approach (based on the Mori scdar product) and the [AA AT] Pad6 approximants. 

A more general expression for g>(t) can be derived from the continued fraction expansion, ref. ... 

Its Laplace transform can be evaluated via a continued fraction expansion (see Chapters I, III, and IV) and is denoted by Note that ... 

We shall set N be the dimension of the finite basis subset used to represent f and v. The calculation can be performed with great efficiency using an iterative algorithm, such as the Lanczos algorithm, that transforms r into a tridiagonalized form. A continued fraction expansion is then obtained ... 

A number of different polynominal series have been proposed in an attempt to develop the best, approximate, simple analytical expression for p jj (E/RT). Blazejowski has recently compared and contrasted their relative efficiency and accuracy. He recommends the use of continued fraction expansion series, since a minimal number of polynomial terms are required to obtain agreement with the exact Vallet values. 

Following the continued fraction expansion formula [1,2], the thermally averaged Green s function matrix G(E) for the effective Hamiltonian in Eq. (1) is given as... 

By using a suitable memory function approach, it is possible to derive an exact expression of the Laplace transform < SxSxiz) > as a continued fraction expansion... 

In this method a new basis set is found, which tridiagonalizes H. The spectrum is then obtained from a continued fraction expansion. The first function in the new basis set is... 

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